As explained in [1]}, considering the closed embedding \(\xi _X:X^0\hookrightarrow X\) , \(X^0\)  admits a natural d-critical structure \((X^0,s^0:=\xi _X^\star (s))\) . Indeed, for \((R,U,f,i)\)  a \(\mathbb {C}^\ast \) -equivariant d-critical chart, denoting by \(R^0\) , \(U^0\)  the \(\mathbb {C}^\ast \) -fixed locus of \(R\) ,\(U\) , and \(f^0=f|_{U^0}\) , \(i^0=i|_{R^0}\) , \((R^0,U^0,f^0,i^0)\)  is a d-critical chart of \(X^0\)  such that \(s^0_{R^0,U^0}=f^0+I^2_{R^0,U^0}\) , and then \(X^0\)  is covered by d-critical charts. Because \(\xi _X:X^0\hookrightarrow X\)  is a closed embedding, \(T^\ast _{X^0/X}=0\) , and then \(K_{X^0,s^0}=\xi _X^\ast (K_{X,s})\)  using [1]}. One can then naturally define the orientation \(K_{X^0,s^0}^{1/2}:=\xi _X^\ast (K_{X,s}^{1/2})\)  on \((X^0,s^0)\) .
